4 THE VAGUE TOPOLOGY 111 4. Let X be a compact space. Show that the topology induced on M+(X) by the vague topology is separable, metrizable and locally compact. (Observe that for each /x0 e M+(X) there exists a neighborhood V of /x0 in M(X) such that V n M+(X) is bounded.) Deduce that the mapping (/x,/)h-»</x,/> of ^c(x) x M+(X) into C is continuous. Deduce that if X is locally compact but not compact, then M+(X) is still metriz- able with respect to the topology induced by the vague topology, but is not separable. 5. (a) Let X be a locally compact space. Show that if (gn) is a sequence of functions in the Frechet space ^CPO tending to 0 and ([*„) is a sequence of measures on X tending vaguely to 0, then the sequence of measures (gn' f^n) tends vaguely to 0. (b) Show that if M+(X) is endowed with the topology induced by the vague topology, the mapping (ju,,#)i-»# • ft of M+(X) x #CP9 into M(X) is continuous. (c) Suppose that X is compact and infinite. Show that the mapping (p,,g)h-+g - p, of M(X) x #C(X) into M(X) is not continuous when M(X) is endowed with the vague topology (argue as in Problem 3(b)). 6. Let X be the unit interval [0,1 ] in R. (a) Let fjLn be the measure e0 — s1/n. Show that the sequence (/xn) tends vaguely to 0 but that the sequence (|ju-n|) tends vaguely to 2e0 - (b) Let A be Lebesgue measure on X, and let gn(x) — sin nx. Show that the sequence of positive measures p,n = (1 — gn) • A converges vaguely to A as n ->- 4- oo, but does not converge to A with respect to the topology defined by the norm (5.7.1) on M(X). 7. Let X be a compact space and /x a positive measure of mass 1 on X. A sequence (xn) of points of X is said to be equirepartitioned with respect to /x if the sequence of measures (n~1(sxl 4- •• • + £*„)) converges vaguely to JLA. Show that a necessary and sufficient condition for this to be the case is that, if (fk) is any total sequence (5.4) in #C(X), then for each k the sequence (n"1(fk(xi) H-----+ /*(*„))) converges to J/* d^. Consider in particular the case where X = [0,1] and xn = n6 — [n6] (where [t] is the integral part of a real number t). Show that if B is irrational, then the sequence (#„) is equirepartitioned with respect to Lebesgue measure ("Bohl's theorem")- (Use the sequence of functions fk(x) = e2nikx.) 8. Let X, Y be two locally compact spaces and TT : X ->• Y a proper continuous mapping (Section 12.7, Problem 2). For each function ge Jf(Y), we have g ° TT e Jf(X). If /x is any measure on X, then the image of /x under TT is defined to be the measure g\-+[ji(go 77) on Y, and is denoted by TT(JLC). The mapping /xi—>TT(/X) of M(X) into M(Y) is linear and vaguely continuous. Suppose that X is compact. Let F be a set of continuous mappings of X into X, each pair of which commute. Let F' be the set of linear mappings of the form /Lch-»/rH/x + w(/x) H-------h un-l(p)) of M(X) into M(X), where u e T and n is any positive integer, and let F" be the set of all compositions of any finite number of elements of Fx. If K is the convex and vaguely compact set of positive measures of total mass 1 on X, then u(K) c K for all u E T". Show that the sets */(K), where u e F7', form a filter base in K consisting of vaguely compact sets, and deduce that the intersection I of these sets is nonempty. Show finally that each measure /x e I is invariant under F, i.e., that v(/x) = /x for all u e F (" MarkofT-Kakutani theorem")- (Observe that, by definition, there exists for each n a measure v e K such that ft — n~~l(v + u(y) + • • • + if~l(v))9 and evaluate ||w(/x) — /x||.) Consider the case of a compact commutative group operating on itself by translation (cf. (14.1)).ting of a single point is closed,